complex integral, partial fractions 2
up vote
1
down vote
favorite
I have to evaluate the following integral:
$ int_{gamma} frac{1}{z^3-z^2+z-1} dz$, $ gamma: [0, 2pi] rightarrow mathbb{C}, gamma(t)=1+e^{2it} $
My idea was to split the integrand into partial fractions and apply cauchys integral formula. Then i get:
$int_{gamma} frac{1}{z^3-z^2+z-1} dz = frac{1}{2}int_{gamma} frac{1}{z-1}dz- frac{1}{4} int_{gamma} frac{1-i}{z-i}dz- frac{1}{4} int_{gamma} frac{1-i}{z+i}dz $
The last 2 integrals are 0, because $ pm i $ arent enclosed by the given contour. For the first one, i get using the cauchy integral formula: $ 2 pi i cdot ind_{Gamma} cdot 2 = int... $. Therefore $int... = 2 pi i cdot 2 cdot 2 = 8 pi i$
Is that right?
integration
asked Nov 25 at 0:06
Sarah34
111
add a comment |
up vote
1
down vote
favorite
I have to evaluate the following integral:
$ int_{gamma} frac{1}{z^3-z^2+z-1} dz$, $ gamma: [0, 2pi] rightarrow mathbb{C}, gamma(t)=1+e^{2it} $
My idea was to split the integrand into partial fractions and apply cauchys integral formula. Then i get:
$int_{gamma} frac{1}{z^3-z^2+z-1} dz = frac{1}{2}int_{gamma} frac{1}{z-1}dz- frac{1}{4} int_{gamma} frac{1-i}{z-i}dz- frac{1}{4} int_{gamma} frac{1-i}{z+i}dz $
The last 2 integrals are 0, because $ pm i $ arent enclosed by the given contour. For the first one, i get using the cauchy integral formula: $ 2 pi i cdot ind_{Gamma} cdot 2 = int... $. Therefore $int... = 2 pi i cdot 2 cdot 2 = 8 pi i$
Is that right?
integration
asked Nov 25 at 0:06
Sarah34
111
Is my solution right?
– Sarah34
Nov 25 at 9:49
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have to evaluate the following integral:
$ int_{gamma} frac{1}{z^3-z^2+z-1} dz$, $ gamma: [0, 2pi] rightarrow mathbb{C}, gamma(t)=1+e^{2it} $
My idea was to split the integrand into partial fractions and apply cauchys integral formula. Then i get:
$int_{gamma} frac{1}{z^3-z^2+z-1} dz = frac{1}{2}int_{gamma} frac{1}{z-1}dz- frac{1}{4} int_{gamma} frac{1-i}{z-i}dz- frac{1}{4} int_{gamma} frac{1-i}{z+i}dz $
The last 2 integrals are 0, because $ pm i $ arent enclosed by the given contour. For the first one, i get using the cauchy integral formula: $ 2 pi i cdot ind_{Gamma} cdot 2 = int... $. Therefore $int... = 2 pi i cdot 2 cdot 2 = 8 pi i$
Is that right?
integration
asked Nov 25 at 0:06
Sarah34
111
I have to evaluate the following integral:
$ int_{gamma} frac{1}{z^3-z^2+z-1} dz$, $ gamma: [0, 2pi] rightarrow mathbb{C}, gamma(t)=1+e^{2it} $
My idea was to split the integrand into partial fractions and apply cauchys integral formula. Then i get:
$int_{gamma} frac{1}{z^3-z^2+z-1} dz = frac{1}{2}int_{gamma} frac{1}{z-1}dz- frac{1}{4} int_{gamma} frac{1-i}{z-i}dz- frac{1}{4} int_{gamma} frac{1-i}{z+i}dz $
The last 2 integrals are 0, because $ pm i $ arent enclosed by the given contour. For the first one, i get using the cauchy integral formula: $ 2 pi i cdot ind_{Gamma} cdot 2 = int... $. Therefore $int... = 2 pi i cdot 2 cdot 2 = 8 pi i$
Is that right?
integration
integration
asked Nov 25 at 0:06
Sarah34
111
asked Nov 25 at 0:06
Sarah34
111
asked Nov 25 at 0:06
Sarah34
111
asked Nov 25 at 0:06
Sarah34
111
asked Nov 25 at 0:06
Sarah34
111
111
Is my solution right?
– Sarah34
Nov 25 at 9:49
add a comment |
Is my solution right?
– Sarah34
Nov 25 at 9:49
Is my solution right?
– Sarah34
Nov 25 at 9:49
Is my solution right?
– Sarah34
Nov 25 at 9:49
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3012266%2fcomplex-integral-partial-fractions-2%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Is my solution right?
– Sarah34
Nov 25 at 9:49